题目
题型:不详难度:来源:
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225135-11518.png)
(1)求证:△AOB的面积为定值;
(2)设直线2x+y-4=0与圆C交于点M、N,若OM=ON,求圆C的方程.
答案
解析
试题分析:(1)先求出圆的方程,然后求出与坐标轴的交点坐标,然后求S△AOB=
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225136-10459.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225136-10459.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225136-90200.png)
试题解析: (1)证明 由题设知,圆C的方程为(x-t)2+
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225136-31626.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225136-50229.png)
化简得x2-2tx+y2-
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225137-84469.png)
当x=0时,y=0或
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225137-84469.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225137-94062.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225136-10459.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225136-10459.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225136-90200.png)
(2)解 ∵OM=ON,则原点O在MN的中垂线上,设MN的中点为H,则CH⊥MN,
∴C、H、O三点共线,则直线OC的斜率k=
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225137-97424.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225137-10296.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225136-10459.png)
∴圆心为C(2,1)或C(-2,-1).
∴圆C的方程为(x-2)2+(y-1)2=5或(x+2)2+(y+1)2=5,
由于当圆方程为(x+2)2+(y+1)2=5时,直线2x+y-4=0到圆心的距离d>r,此时不满足直线与圆相交,故舍去.
∴圆C的方程为(x-2)2+(y-1)2=5.
核心考点
试题【已知以点C (t∈R,t≠0)为圆心的圆与x轴交于点O、A,与y轴交于点O、B,其中O为原点.(1)求证:△AOB的面积为定值;(2)设直线2x+y-4=0与圆】;主要考察你对点到直线的距离等知识点的理解。[详细]
举一反三
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225114-12878.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225114-37813.png)
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![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225114-60864.png)
(Ⅰ)求
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225114-21475.png)
(Ⅱ)以OA,OB为邻边作平行四边形OADB,是否存在常数
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225114-21475.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225114-21475.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225105-35884.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225105-59444.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225105-70495.png)
A.![]() | B.![]() | C.3 | D.![]() |
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225056-11403.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225056-88172.png)
A.![]() | B.![]() | C.![]() | D.![]() |
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225050-53751.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225050-22278.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225050-28575.png)
(Ⅰ)求
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225050-30895.png)
(Ⅱ)当
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225050-17387.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225044-71538.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225045-72426.png)
![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225045-56571.png)
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![](/home/wwwroot/www.shitiku.com.cn/file/uploads/allimg/20191022/20191022225045-24683.png)
A.相交且过圆心 | B.相交但不过圆心 | C.相切 | D.相离 |
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